INITIAL CONDITIONS JING LIU∗ AND MARK FRENCH∗
Abstract. We consider the development of a general nonlinear input-output theory which
encompasses systems with initial conditions. Systems are defined in a set theoretic manner from input-output pairs on a doubly infinite time axis, and a general construction of the initial conditions is given in terms of an equivalence class of trajectories on the negative time axis. Input-output operators are then defined for given initial conditions, and a suitable notion of input-output stability on the positive time axis with initial conditions is given. This notion of stability is closely related to the ISS/IOS concepts of Sontag. A fundamental robust stability theorem is derived which represents a generalization of the input-output operator robust stability theorem of Georgiou and Smith, to include the case of initial conditions. This includes a suitable generalization of the nonlinear gap metric. Some applications are given to show the utility of the robust stability theorem.
Key words. nonlinear systems, robust stability, gap metric, feedback connection, small-gain-like
stability theorem, ISS/IOS
AMS subject classifications. 93C10, 93D09, 93D25, 93D15, 93D20
1. Introduction. The general nonlinear input-output theory of systems was ini-tiated in the 1960s by Zames [42, 43] and Sandberg [23, 24]. It views systems as black boxes identified with operators mapping inputs to outputs. Such approaches, includ-ing recent contributions such as [13], do not include a systematic treatment of initial conditions. On the other hand, states and initial conditions have been introduced into input-output reasoning via the well-known input-to-state stability (ISS) theory due to Sontag [25] (and it’s many variants, see e.g., [27, 28, 32]). The ISS approach is fundamentally a state space approach in which systems are assumed to have a known state space representation (and e.g., Lyapunov constructions and characterizations play a significant role). In contrast input/output approaches to robust stability are concerned with perturbations to nominal systems which induce significant (and poten-tially unknown) changes to the underlying state space. For example, whilst a nominal system may be modelled by a low order finite dimensional system, the true system is viewed as a perturbed system typically with a differing dimension, as occurs with a finite dimensional multiplicative perturbation, or represents a shift from a nominal model with a finite dimensional state space to an infinite dimensional system. For example, consider the nominal plant Σ with one dimensional state space:
Σ : ˙𝑥(𝑡) = 𝜙(𝑥(𝑡)) + 𝑢(𝑡), 𝑦(𝑡) = 𝑥(𝑡), and the perturbed plant Σ𝜏 with infinite dimensional state space
Σ𝜏 : ˙𝑥(𝑡) = 𝜙(𝑥(𝑡)) + 𝑢(𝑡 − 𝜏), 𝑦(𝑡) = 𝑥(𝑡), 0 < 𝜏 ≤ 𝜏0,
where 𝜙 : ℝ → ℝ is a memoryless nonlinear function satisfying a sector condition (Example 4.3). The plants Σ and Σ𝜏are close in the sense of nonlinear gap metric [13]
(𝛿(Σ, Σ𝜏) → 0 as 𝜏 → 0), but with different dimensional state spaces, and one would
anticipate that a satisfactory feedback controller for Σ will also work for Σ𝜏 for any
0 < 𝜏 ≤ 𝜏0provided 𝜏0is sufficiently small. The initial condition in Σ can be taken to ∗School of Electronics and Computer Science, University of Southampton, Southampton,
SO17 1BJ, UK ([email protected], [email protected]). 1
be 𝑥(0) ∈ ℝ. However, for Σ𝜏 the initial condition is necessarily infinite dimensional,
e.g., (𝑥(0), 𝑢∣(−𝜏,0]) ∈ ℝ×𝐿2(−𝜏, 0]. Intuitively, even when initial conditions are taken
into consideration, the nominal plant Σ when stabilized by a controller should remain stabilized when replaced by any of the perturbed plants Σ𝜏, 0 < 𝜏 ≤ 𝜏0. Clearly
to quantify such statements, we need appropriate notions of stability together with an appropriate quantification of the notion of ‘size’ of initial conditions which can be consistently applied across Σ and Σ𝜏 for any 0 < 𝜏 ≤ 𝜏0. Additionally these concepts
must also be applicable to all other ‘reasonable’ perturbations (multiplicative, additive etc.) which often change the state space structure, and move the scope of the required framework beyond that of state space representations.
The purpose of this paper is to develop a general input/output framework which incorporates a general concept of initial conditions. The central result obtained is a generalization of the robust stability results of [13] whereby the initial conditions (characterized by a purely input-output formalism drawn from [38]) are reflected within the stability concept in an ISS-like manner (cf. [25, 27, 28, 32]). We remark that a central assumption in [13] is a requirement that systems are defined on semi-infinite time axes and map zero inputs onto zero outputs. Implicitly this requires that the systems have zero initial conditions. There are a number of later extensions which permit consideration of nonzero responses to zero disturbances, e.g., [8, 12], however, neither of these approaches are directly aimed at the case of initial conditions, and cannot directly be used to establish fading memory properties. Explicit robust stability results are given in [7, 9] for a specific case of a linear plant and a nonlinear controller with initial conditions. A more general construction for nonlinear plants can be found in [10, §7], and this forms the basis for this contribution.
Thus, on the one hand the contribution of this paper can be viewed as a gen-eralization of the ISS approach to enable a realistic treatment of robust stability in the context of perturbations which fundamentally change the structure of the state space; and on the other hand can be viewed as a generalization of existing operator based input-output approaches to robust stability to include initial conditions within, in particular, the nonlinear gap formalism of [13].
This paper is organized as follows. In §2, we introduce definitions of systems, ini-tial conditions and closed-loop systems, which involve only input-output structures. In
§2 we show how the general initial condition construction relates to standard notions
of initial conditions for systems having particular representations. The fundamental robust stability theorem for input-output feedback systems with initial conditions in terms of a generalized gap metric is given in §3. Section 4 considers some applications to show the effects of this paper’s results. We draw conclusions in §5.
2. Systems, initial conditions and closed-loop systems. Let 𝒮 denote the set of all locally integrable maps ℝ → 𝒳 where 𝒳 is a nonempty set. For any interval
𝐽, we regard 𝒮𝐽 as a subspace of 𝒮 by identifying 𝒮𝐽 with the set of maps in 𝒮 which
vanish outside of 𝐽. We define a truncation operator 𝑇𝐽 : 𝒮 → 𝒮 and a restriction
operator 𝑅𝐽: 𝒮𝐼 → 𝒮𝐽 with 𝐽 ⊂ 𝐼 as follows:
𝑇𝐽 : 𝒮 → 𝒮, 𝑣 7→ 𝑇𝐽𝑣 ≜ ( 𝑡 7→ { 𝑣(𝑡), 𝑡 ∈ 𝐽 0, otherwise ) ; 𝑅𝐽 : 𝒮𝐼 → 𝒮𝐽, 𝑣 7→ 𝑅𝐽𝑣 ≜ ( 𝑡 7→ 𝑣(𝑡), 𝑡 ∈ 𝐽).
We let 𝑅+ ≜ 𝑅[0,∞) and 𝑅− ≜ 𝑅(−∞,0]. For any 𝑢, 𝑣 ∈ 𝒮 and any 𝜏 ∈ ℝ, the 𝜏-concatenation of 𝑢 and 𝑣, denoted 𝑢 ∧𝜏𝑣, is defined by: (𝑢 ∧𝜏𝑣)(𝑡) = 𝑢(𝑡) if 𝑡 ≤ 𝜏;
and (𝑢 ∧𝜏𝑣)(𝑡) = 𝑣(𝑡) if 𝑡 > 𝜏. We abbreviate 𝑢 ∧ 𝑣 ≜ 𝑢 ∧0𝑣. Define 𝒱 ⊆ 𝒮 to be
a signal space if and only if it is a vector space. Suppose additionally that 𝒱 is a normed vector space and the norm ∥ ⋅ ∥ = ∥ ⋅ ∥𝒱is also defined for signals of the form
𝑇𝐽𝑣, 𝑣 ∈ 𝒱, 𝐽 ⊆ ℝ. We define a norm ∥⋅ ∥𝐽 on 𝒮𝐽by ∥𝑣∥𝐽 = ∥𝑇𝐽𝑣∥ for 𝑣 ∈ 𝒮𝐽 (define
∥𝑣∥𝐽≜ ∞ if 𝑇𝐽𝑣 ∈ 𝒮 ∖ 𝒱). The extended space 𝒱𝑒 of 𝒱 is defined by
𝒱𝑒≜{𝑣 ∈ 𝒮 ∣ ∀𝑎, 𝑏, (−∞ < 𝑎 < 𝑏 < ∞) : 𝑇(𝑎,𝑏)𝑣 ∈ 𝒱},
and the interval space 𝒱(𝐽) ≜ 𝑅𝐽𝒱 for any 𝐽 ⊂ ℝ; we also abbreviate 𝒱+ = 𝑅+𝒱, 𝒱− = 𝑅−𝒱, 𝒱+
𝑒 = 𝑅+𝒱𝑒 and 𝒱𝑒− = 𝑅−𝒱𝑒. In the rest of this paper, unless
speci-fied otherwise, we always let 𝒰, 𝒴 be normed (input/output) signal spaces (such as
𝐿𝑞(ℝ; ℝ𝑛), 1 ≤ 𝑞 ≤ ∞) with norm ∥ ⋅ ∥
𝒰, ∥ ⋅ ∥𝒴, respectively. Let 𝒲 ≜ 𝒰 × 𝒴 with
the product norm defined in the usual way, ∥(𝑢, 𝑦)∥𝒲 = (∥𝑢∥𝑞𝒰 + ∥𝑦∥𝑞𝒴)
1
𝑞 if 𝑞 ≥ 1,
∥(𝑢, 𝑦)∥𝒲= max {∥𝑢∥𝒰, ∥𝑦∥𝒴} if 𝑞 = ∞.
2.1. Systems. In an input/output framework it is only the relationship between inputs and outputs that is a-priori relevant. In this sense, notions of ‘system’ and of ‘stability’ should be made without the axiomatical postulation of state.
Definition 2.1. Given normed signal spaces 𝒰, 𝒴 and 𝒲 ≜ 𝒰 × 𝒴, a system 𝑄
is defined via the specification of a subset 𝔅𝑄 ⊂ 𝒲𝑒.
The signal pair (𝑢, 𝑦) ∈ 𝒰𝑒× 𝒴𝑒 is called an input-output pair. At this stage,
we do not impose any further requirements on the input/output partition. In an operator based input/output framework (e.g., [13]), it would be typical to start with an operator 𝑄: 𝒰𝑒→ 𝒴𝑒, where e.g., 𝒰𝑒= 𝒴𝑒≜ 𝐿2𝑒(ℝ+; ℝ) and 𝑄(0) = 0. Note that
the above definition of a system differs from both Zames’s representation of input-output systems by operators [41] and Willems’s structure of input-input-output systems by behaviors with input/output partition [21]. Here, we allow both (𝑢, 𝑦1) and (𝑢, 𝑦2)
with 𝑦1∕= 𝑦2belong to the same set 𝔅𝑄. And it does not require that for any 𝑢 ∈ 𝒰𝑒
there exists a 𝑦 ∈ 𝒴𝑒such that (𝑢, 𝑦) ∈ 𝔅𝑄. For example, Let 𝒰 = 𝒴 ≜ 𝐿2(ℝ; ℝ) and
consider the system 𝑄 represented by the set 𝔅𝑄={(𝑢, 𝑦) ∈ 𝒰𝑒× 𝒴𝑒∣ 𝑦2= 𝑢}. It is
easy to verify that for 𝑢(𝑡) = 𝑒−2∣𝑡∣, 𝑡 ∈ ℝ and 𝑦(𝑡) = 𝑒−∣𝑡∣, 𝑡 ∈ ℝ we have both (𝑢, 𝑦)
and (𝑢, −𝑦) belong to 𝔅𝑄, and that for 𝑢(𝑡) = −𝑒−2∣𝑡∣, 𝑡 ∈ ℝ, there is no 𝑦 ∈ 𝐿2𝑒(ℝ; ℝ)
such that (𝑢, 𝑦) ∈ 𝔅𝑄. Since our set 𝔅𝑄 allows us to consider multi-valued maps
𝑄 or relations 𝑄, we will see in subsequent sections that this is key to our unified
treatment of initial conditions.
Definition 2.2. A system 𝑄 is said to be linear if the set 𝔅𝑄 is a vector space,
i.e., 𝜆1𝑤1+ 𝜆2𝑤2 ∈ 𝔅𝑄 for any 𝑤1, 𝑤2 ∈ 𝔅𝑄 and any 𝜆1, 𝜆2 ∈ ℝ. It is said to be
time-invariant if 𝑤 ∈ 𝔅𝑄 implies 𝑤(⋅ + 𝜏) ∈ 𝔅𝑄 for all 𝜏 ∈ ℝ.
Definition 2.3. Given normed signal spaces 𝒰 and 𝒴, an operator Φ : 𝒰+
𝑒 → 𝒴𝑒+
is said to be causal if, ∀𝑢, 𝑣 ∈ 𝒰+
𝑒 , ∀𝑡 > 0 :
[
𝑢∣[0,𝑡]= 𝑣∣[0,𝑡]⇒ (Φ𝑢)∣[0,𝑡]= (Φ𝑣)∣[0,𝑡]], while a system 𝑄 is said to be causal if
∀𝑢, 𝑣 ∈ 𝒰𝑒, ∀𝑡 ∈ ℝ :[𝑢∣(−∞,𝑡]= 𝑣∣(−∞,𝑡]⇒ 𝔅𝑄𝑢∣(−∞,𝑡]= 𝔅𝑣𝑄∣(−∞,𝑡]], where 𝔅𝑢
𝑄 ≜
{
𝑤 ∈ 𝒲𝑒 ∃𝑦 ∈ 𝒴𝑒 such that 𝑤 = (𝑢, 𝑦) ∈ 𝔅𝑄}.
This definition generalizes the definition of a casual operator. Note that any operator Φ : 𝒰+
𝑒 → 𝒴𝑒+can be represented by a system 𝔅Φ= {(𝑢, 𝑦) ∈ 𝒰𝑒×𝒴𝑒∣ 𝑅−𝑦 =
if and only if the system 𝔅Φ is causal. We will be interested to define system’s
properties using trajectories defined on the positive half-line [𝑡, ∞). In order to define the well-posedness of a system, we first introduce the two properties of existence and uniqueness of a system. In the following, we fixed initial time 𝑡 = 0 if not otherwise specified and use the notation 𝔅−
𝑄 defined as follows to denote the system 𝑄’s past
trajectories: 𝔅−
𝑄≜ 𝑅−𝔅𝑄 ={𝑤−∈ 𝒲𝑒− ∣ ∃ 𝑤+∈ 𝒲𝑒+, s.t. 𝑤−∧𝑤+∈ 𝔅𝑄}. (2.1)
Definition 2.4. A system 𝑄 is said to have the existence property if for any
𝑤− ∈ 𝔅−𝑄 and any 𝑢+ ∈ 𝒰𝑒+ there exists a 𝑦+ ∈ 𝒴𝑒+ such that 𝑤−∧(𝑢+, 𝑦+) ∈ 𝔅𝑄;
and the uniqueness property if for any 𝑤− ∈ 𝔅−𝑄 and any 𝑢+∈ 𝒰𝑒+,
𝑤−∧(𝑢+, 𝑦+), 𝑤−∧(𝑢+, ˜𝑦+) ∈ 𝔅𝑄 with 𝑦+, ˜𝑦+∈ 𝒴𝑒+ ⇒ 𝑦+= ˜𝑦+, and is well-posed if it has both the existence and uniqueness properties.
Well-posedness means that future output 𝑦+ can be deduced from the set 𝔅𝑄
(representing system properties), the past input-output pair (𝑢−, 𝑦−) and the future
input 𝑢+ and is equivalent to the concept that output processes input as defined in
[39]. The graph 𝒢𝑄𝑤− of a system 𝑄 for a given past trajectory 𝑤−∈ 𝔅−𝑄is defined by
𝒢𝑄𝑤−≜{𝑤+ ∈ 𝒲+ ∣ 𝑤−∧𝑤+∈ 𝔅𝑄}⊂ 𝒲+. (2.2)
2.2. Initial conditions. As discussed in intuitive terms in the control literature, see [40, Chapter 1] and [42], the state is a classifier of input-output pasts and the state should contain all the information of past history of the system which at any time together with the future input completely determine the future output. The state at time 0 thus determines the initial conditions. In the following, we will give a precise way to define the state of an arbitrary input/output system. It is fundamental that the construction does not require a system representation, but we do show how the construction relates to the standard concepts of state for significant classes of system representations. The genesis of this approach lies in [10, §7]. From the viewpoint of observability, for any observable nonlinear system represented by a state space model, the initial state can be reconstructed from observed output signals given some known input signals (see e.g., [11]).
We now define an equivalence relation on 𝔅−𝑄≜ 𝑅−𝔅𝑄 (see (2.1)) as follows: for
any 𝑤−, ˜𝑤− ∈ 𝔅−𝑄, we say
𝑤− ∼ ˜𝑤− ⇔ 𝑄𝑤−(𝑢+) = 𝑄𝑤−˜ (𝑢+), ∀𝑢+∈ 𝒰𝑒+, (2.3)
where 𝑄𝑤−(𝑢
+) denotes the set (possibly empty) of all future outputs generated by
the system past input-output 𝑤− ∈ 𝔅−𝑄 and future input 𝑢+∈ 𝒰𝑒+, i.e.,
𝑄𝑤−(𝑢
+) ≜{𝑦+∈ 𝒴𝑒+ ∣ 𝑤−∧(𝑢+, 𝑦+) ∈ 𝔅𝑄}. (2.4)
The equivalence class of any 𝑤− in 𝔅−𝑄 is denoted by [𝑤−] ≜ { ˜𝑤−∈ 𝔅−𝑄 ∣ ˜𝑤−∼ 𝑤−}.
Definition 2.5. We define 𝔖𝑄 the initial state space of 𝑄 at initial time 0 as
the quotient set 𝔅−
𝑄/ ∼, i.e., 𝔖𝑄 = 𝔅−𝑄/ ∼ ≜ {[𝑤−] ∣ 𝑤−∈ 𝔅−𝑄}.
From the equivalence relation ∼, for any 𝑥0∈ 𝔖𝑄, we define the set 𝑄𝑥0(𝑢+) by: 𝑄𝑥0(𝑢
If the initial time is chosen to be 𝑡0∈ ℝ, we can similarly define the initial state space
denoted by 𝔖𝑡0
𝑄 of a system 𝑄 at initial time 𝑡0 by the same procedure. Note that
the above definition of initial state space doesn’t require the system to be well-posed; however, if so, then, there is a unique element in 𝑄𝑤−(𝑢
+) for every 𝑤− ∈ 𝔅−𝑄 and
every 𝑢+∈ 𝒰𝑒+; and in this case, 𝑄𝑤−(⋅) can be regarded as an operator from 𝒰𝑒+ to
𝒴+
𝑒 for every 𝑤− ∈ 𝔅−𝑄. In turn, this implies that for every 𝑥0 ∈ 𝔖𝑄, 𝑄𝑥0(⋅) is an
operator from 𝒰+
𝑒 to 𝒴𝑒+.
This equivalence class construction of the initial state space is not new; it is closely related to the construction of states in automata (or machine) theory and control theory via Nerode equivalence appearing in slightly different manner. This technique was introduced by Nerode [20] when defining a state-equivalence relation in linear automata theory. The formal definition of Nerode equivalence can be found in [22, p. 114] in the generale setting of automata theory including the nonlinear case; in [3] for discrete-time systems from an abstract algebraic point of view; in [18, Chapters 7 and 10] including a discussion of connection between automata and control theory; and in [30, p. 309] for any time-invariant input/output behaviors including both discrete-time and continuous-time cases. The equivalence relation considered in this paper is slightly different from the one considered in [30, p. 309] where equivalence classes only relate to input sequences, since we do not restrict ourself to input/output behaviors which can be associated with an input/output map, hence the equivalence class is build from both input and output pairs.
Within the behavioral approach, Willems constructs three canonical state repre-sentations by introducing three equivalence relations for a given system represented by a behavior [38]. The construction of state in this paper is similar to the
past-induced canonical state representation in [38]. Note that in this paper we do not
impose any requirements on the input/output partition for a system (see Definition 2.1). This construction of state enables us to define the well-posedness of a system and a closed-loop system in a unified way (see below). Notice that this is different from giving a definition of well-posedness for a system with Willems’ input/output partition [21, Definition 3.3.1]; since any systems with Willems’ input/output parti-tion already guarantee the existence property which is a very important property of a closed-loop system. Hence we relax the requirement that the input is free in [21, Definition 3.3.1] in order to study closed-loop systems.
A functional 𝜒 assigns a notion of size to elements in the initial state space 𝔖𝑄:
𝜒 : 𝔖𝑄→ [0, ∞], 𝑥07→ 𝜒(𝑥0) ≜ inf {∥𝑤−∥ ∣ 𝑤− ∈ 𝑥0} . (2.6)
This notion of size defined above related to finite energy reachability may be inter-preted as the minimization of energy of the past system trajectories that ‘explain’ the corresponding initial state. Notice that in §3.2 we will give a detailed discussion about the concept of finite-time reachability which roughly means that any state can be reached from zero state by finite time. The notion of size defined above may also be interpreted as the required supply in the context of dissipative dynamical systems, see e.g., [37]. The determination of 𝜒 is a standard problem in optimal control, see e.g., [1]. It is well known that, for 𝐿2 norm (square-integrable) in the linear case
˙𝑥 = 𝐴𝑥 + 𝐵𝑢, 𝑦 = 𝐶𝑥 + 𝐷𝑢, above infimum is simplified to the traditional linear quadratic optimal control problem with ∥𝑤−∥1/2 = ∫−∞0 [𝑢𝑇(𝑡)𝑢(𝑡) + 𝑦𝑇(𝑡)𝑦(𝑡)] 𝑑𝑡.
Moreover, if the considered linear system is minimal (i.e., (𝐴, 𝐵) controllable and (𝐴, 𝐶) observable) and 𝐷 + 𝐷𝑇 is invertible, then there exists a real symmetric
is one-to-one related to 𝑥0∈ 𝔖𝑄 by the bijection obtained from Corollary 2.10.
We conclude this section by giving some properties of 𝔖𝑄 and 𝜒 for the linear
systems:
Proposition 2.6. If the system 𝑄 is linear, then the initial state space 𝔖𝑄 is a
vector space. Moreover, the functional 𝜒 given by (2.6) defines a norm on 𝔖𝑄.
Proof. It is elementary to show that the initial state space 𝔖𝑄 is a vector space
with 0 =[0∣(−∞,0]]as its additive identify and satisfies 𝜒(0) = 0. From the definition
of 𝜒 (see (2.6)), it is easy to see that 𝜒(𝑧0) ≥ 0 for any 𝑧0∈ 𝔖𝑄 and that if 𝜒(𝑧0) = 0,
then we must have 0∣(−∞,0] ∈ 𝑧0 (i.e., 𝑧0 = 0). For any 𝑥0 = [𝑤−] ∈ 𝔖𝑄 and
any 𝜆 ∈ ℝ, we have 𝜒(𝜆 ⋅ 𝑥0) = 𝜒([𝜆 ⋅ 𝑤−]) = ∣𝜆∣ 𝜒([𝑤−]) = ∣𝜆∣ 𝜒(𝑥0). Since for
any 𝑤1−, 𝑤2− ∈ 𝔅−𝑄 we have ∥𝑤1−+ 𝑤2−∥ ≤ ∥𝑤1−∥ + ∥𝑤2−∥, and thus we obtain 𝜒(𝑥0+ 𝑦0) ≤ 𝜒(𝑥0) + 𝜒(𝑦0) for any 𝑥0, 𝑦0∈ 𝔖𝑄. Therefore 𝜒 defines a norm on 𝔖𝑄
for any linear system 𝑄.
2.2.1. The relation to state space initial conditions. Consider a system Σ described by the following state-space model:
˙𝑥 = 𝑓(𝑥, 𝑢), 𝑦 = ℎ(𝑥, 𝑢), (2.7) where 𝑢(𝑡) ∈ ℝ𝑚 (𝑡 ∈ ℝ) is the input variable, and 𝑥(𝑡) ∈ 𝑀 ⊂ ℝ𝑙 denotes the state
variable (𝑀 is an open set) and 𝑦(𝑡) ∈ ℝ𝑝 represents the output variable, and both
𝑓 : 𝑀 × ℝ𝑚 → 𝑀 and 𝑔 : 𝑀 × ℝ𝑚 → ℝ𝑝 are continuous functions. Define signal
spaces 𝒰 ≜ 𝐿𝑞(ℝ; ℝ𝑚) (1 ≤ 𝑞 ≤ ∞), 𝒴 ≜ 𝐿𝑞(ℝ; ℝ𝑝) (1 ≤ 𝑞 ≤ ∞) and 𝒲 ≜ 𝒰 × 𝒴.
According to Definition 2.1, the system Σ is defined by the set:
𝔅Σ= {𝑤 ∈ 𝒲𝑒∣ 𝑤 = (𝑢, 𝑦) and (2.7) satisfies for some 𝑥(𝑡) ∈ 𝑀(𝑡 ∈ ℝ)} . (2.8)
By using the same procedure in §2.2, we can define the initial state space 𝔖Σfor the
above set 𝔅Σat initial time 0.
Definition 2.7. The state space model (2.7) is said to be forward complete [2],
if for any 𝑢+∈ 𝒰𝑒+ and any initial state 𝑥0∈ 𝑀, there exists a unique 𝑥(𝑡) ∈ 𝑀 (for all 𝑡 ≥ 0) satisfying (2.7). It is said to be backward complete, if for every 𝑢− ∈ 𝒰𝑒−
and every initial state 𝑥0, there exists a unique 𝑥(𝑡) ∈ 𝑀 (for all 𝑡 ≤ 0) satisfying (2.7). It is said to be complete if it is both forward complete and backward complete.
It is well-known that the state space model (2.7) is complete if 𝑓 is continuous in 𝑡 and 𝑢 and Lipschitz continuous in 𝑥 (see e.g., [4]). Suppose that the state space model (2.7) is a complete representation. If the trajectories of (2.7) are required to satisfy the initial condition 𝑥(0) = 𝑥0 (𝑥0∈ 𝑀), then the state space model defines
a forward operator Σ𝑥0
+ from 𝒰𝑒+ to 𝒴𝑒+ as follows: each input 𝑢+∈ 𝒰𝑒+ gives rise to
a solution 𝑥(𝑡) ∈ 𝑀 (𝑡 ≥ 0) of ˙𝑥 = 𝑓(𝑥, 𝑢) satisfying the initial condition 𝑥(0) = 𝑥0.
This in turn defines an output 𝑦+∈ 𝒴𝑒+ by 𝑦+(𝑡) = ℎ(𝑥(𝑡), 𝑢+(𝑡)) (𝑡 ≥ 0), i.e.,
Σ𝑥0
+ : 𝒰𝑒+→ 𝒴𝑒+, 𝑢+7→ 𝑦+. (2.9)
A backward operator Σ𝑥0
− : 𝒰𝑒−→ 𝒴𝑒− can be similarly defined like (2.9).
Definition 2.8. Suppose that the state space model (2.7) is complete. It is said
to be forward observable if (see e.g., [14]), for any initial states 𝑥0, 𝑥′0 ∈ 𝑀 with 𝑥0∕= 𝑥′0, there exists some 𝑢+ ∈ 𝒰𝑒+ such that Σ𝑥+0(𝑢+) ∕= Σ𝑥
′
0
+(𝑢+). It is said to be
strongly forward observable if, for any initial states 𝑥0, 𝑥′0 ∈ 𝑀 with 𝑥0 ∕= 𝑥′0, for any 𝑢+∈ 𝒰𝑒+, we have Σ𝑥+0(𝑢+) ∕= Σ𝑥
′
0
+(𝑢+). It is said to be backward observable if, for any initial states 𝑥0, 𝑥′0 ∈ 𝑀 with 𝑥0∕= 𝑥′0, there exist some 𝑢− ∈ 𝒰𝑒− such that
Σ𝑥0
−(𝑢−) ∕= Σ𝑥 ′
0
−(𝑢−). It is said to be strongly backward observable if, for any initial
states 𝑥0, 𝑥′0∈ 𝑀 with 𝑥0∕= 𝑥′0, for any 𝑢−∈ 𝒰𝑒−, we have Σ𝑥−0(𝑢−) ∕= Σ𝑥 ′
0
−(𝑢−).
We let 𝔅−
Σ(𝑥0) denote the set of all past input-output trajectories which are
compatible with the initial state 𝑥0∈ 𝑀 at initial time 0:
𝔅− Σ(𝑥0) ≜ {( 𝑢− 𝑦− ) 𝑢−∈ 𝒰𝑒−, 𝑦− ∈ 𝒴𝑒− and (2.7) satisfies
for some 𝑥(𝑡) ∈ 𝑀(𝑡 ≤ 0) with 𝑥(0) = 𝑥0
}
. (2.10) Proposition 2.9. Suppose that the state space model (2.7) is complete,
for-ward observable and strongly backfor-ward observable. Then 𝐹 : 𝑥07→ 𝔅−Σ(𝑥0) defines a bijection from 𝑀 to 𝔖Σ.
Proof. From Definition 2.5, the initial state space at time 0 of 𝔅Σ is defined
by 𝔖Σ ≜ 𝔅−Σ/ ∼ with 𝔅−Σ ≜ 𝑅−𝔅Σ (see (2.1)), and the corresponding equivalence
relation ∼ on 𝔅−
Σ is defined as follows (see (2.4) and (2.3)): for any 𝑤−, ˜𝑤−∈ 𝔅−Σ, 𝑤−∼ ˜𝑤− ⇔ Σ𝑤−(𝑢+) = Σ𝑤−˜ (𝑢+), ∀𝑢+∈ 𝒰𝑒+. (2.11)
We obtain from (2.8) and (2.10) that 𝔅−
Σ =
∪
𝑥0∈𝑀
{ 𝔅−
Σ(𝑥0)}. Since the state
space model (2.7) is complete and strongly backward observable, we have 𝔅−
Σ(𝑥0) ∩
𝔅−
Σ(𝑥′0) = ∅ for any 𝑥0, 𝑥′0∈ 𝑀 with 𝑥0∕= 𝑥′0. In addition, for any 𝑤−∈ 𝔅−Σ(𝑥0) and
any 𝑢+ ∈ 𝒰𝑒+, we have Σ𝑤−(𝑢+) = Σ+𝑥0(𝑢+) with Σ𝑥+0(𝑢+) defined by (2.9). Thus,
for any 𝑥0∈ 𝑀, the set 𝔅−Σ(𝑥0) is a subset of some equivalence class related to the
equivalence relation ∼. Since the state space model (2.7) is also forward observable, (i.e., Σ𝑥0
+ ∕= Σ𝑥
′
0
+, ∀𝑥0, 𝑥′0∈ 𝑀 with 𝑥0 ∕= 𝑥′0), we get that 𝔅−Σ(𝑥0) and 𝔅−Σ(𝑥′0) must
be contained in two different equivalence classes related to the equivalence relation ∼. This, in turn, implies that{𝔅−Σ(𝑥0) ∣ 𝑥0∈ 𝑀}is the exact partition1 of 𝔅−Σ related
to the equivalence relation ∼. Therefore, we have 𝔖Σ= {𝔅−Σ(𝑥0) ∣ 𝑥0∈ 𝑀} and the
map 𝐹 : 𝑥07→ 𝔅−Σ(𝑥0) is a bijection from 𝑀 to 𝔖Σ.
Corollary 2.10. If the system Σ defined by (2.7) is a linear time invariant
(LTI) system, i.e., ˙𝑥 = 𝑓(𝑥, 𝑢) = 𝐴𝑥 + 𝐵𝑢 and 𝑦 = ℎ(𝑥, 𝑢) = 𝐶𝑥 + 𝐷𝑢, where 𝑥(𝑡) ∈ 𝑀 = ℝ𝑛, 𝑢(𝑡) ∈ ℝ𝑚 and 𝑦(𝑡) ∈ ℝ𝑝 for any 𝑡 ∈ ℝ, and 𝐴, 𝐵, 𝐶, 𝐷 are
appropriate dimensional matrixes. Suppose that (𝐴, 𝐶) is observable [45], i.e., the 𝑛𝑝 × 𝑛 observability matrix [𝐶𝑇, (𝐶𝐴)𝑇, ⋅ ⋅ ⋅ , (𝐶𝐴𝑛−1)𝑇]𝑇 are of full column rank 𝑛.
Then there exists a bijective map from 𝑀 = ℝ𝑛 to 𝔖
Σ.
Proof. Since 𝑓(𝑥, 𝑢) = 𝐴𝑥 + 𝐵𝑢 is continuous in 𝑢 and Lipschitz continuous in 𝑥, this implies that Σ is complete. While for LTI system the observability matrix
[𝐶𝑇, (𝐶𝐴)𝑇, ⋅ ⋅ ⋅ , (𝐶𝐴𝑛−1)𝑇]𝑇 has full column rank 𝑛 implies that the system is
for-ward observable and strongly backfor-ward observable. Thus from Proposition 2.9 there exists a bijective map from 𝑀 = ℝ𝑛 to 𝔖
Σ.
2.2.2. Initial conditions for a concrete delay line. To give a further insight into the abstract notion of initial conditions for systems, consider the time delay line model where the output signal is simply a time delayed copy of the input signal. Define input and output signal spaces 𝒰 = 𝒴 = 𝐿∞(ℝ; ℝ) and 𝒲 ≜ 𝒰 × 𝒴. Then the
input-output system of the time 𝜏-delay line model is:
𝔅𝜏≜ {(𝑢, 𝑦) ∈ 𝒲𝑒 ∣ 𝑦(𝑡) = 𝑢(𝑡 − 𝜏), ∀𝑡 ∈ ℝ} . (2.12) 1Given any set 𝑋, let 𝑁 be a subsets of 𝑋. Then 𝑁 is called a partition of 𝑋 if, and only if, the
According to (2.1), the set of past trajectories 𝔅− 𝜏 is defined by 𝔅− 𝜏 = { (𝑢−, 𝑦−) ∈ 𝒲𝑒− ∣ 𝑦−(𝑡) = 𝑢−(𝑡 − 𝜏), ∀𝑡 ≤ 0}. (2.13)
According to Definition 2.5, the initial state space of 𝔅𝜏 is the quotient set 𝔅−𝜏/ ∼
with the equivalence relation ∼ on 𝔅−
𝜏 defined by
𝑤− ∼ ˜𝑤−⇔ 𝑢−(𝑡) = ˜𝑢−(𝑡), ∀𝑡 ∈ [−𝜏, 0], (2.14)
where 𝑤−= (𝑢−, 𝑦−) ∈ 𝔅−𝜏 and ˜𝑤−= (˜𝑢−, ˜𝑦−) ∈ 𝔅−𝜏. And the equivalent class [𝑤−]
of any element 𝑤−= (𝑢−, 𝑦−) ∈ 𝔅−𝜏 is
[𝑤−] ={(˜𝑢−, ˜𝑦−) ∈ 𝔅−𝜏 ∣ ˜𝑢−∣[−𝜏,0]= 𝑢−∣[−𝜏,0]}. (2.15)
The real-valued function 𝜒 on 𝔅−
𝜏/ ∼ is defined by
[𝑤−] 7→ 𝜒([𝑤−]) ≜ inf {∥ ˜𝑤−∥ ∣ ˜𝑤− ∈ [𝑤−]} .
According to (2.5) and (2.4), let 𝑠0 ∈ 𝔅−𝜏/ ∼ be any initial state of 𝔅𝜏, and let
𝑢+ ∈ 𝒰𝑒+ denote the future input signal of 𝔅𝜏, and let 𝑦+ ∈ 𝒴𝑒+ denote the future
output signal of 𝔅𝜏, then we have
𝑦+(𝑡) = (𝑄𝑠𝜏0(𝑢+))(𝑡) ≜
{
𝑢−(𝑡 − 𝜏), for 𝑡 ∈ [0, 𝜏],
𝑢+(𝑡 − 𝜏), for 𝑡 > 𝜏, (2.16)
where 𝑤−= (𝑢−, 𝑦−) ∈ 𝔅−𝜏 is any element in 𝑠0.
We know that the time 𝜏-delay line model is an abstract linear system, i.e., a quadruple (𝕋, Φ, Ψ, 𝔽) defined in Weiss [34, p. 831]. Let the classical state space be
𝑋 = 𝐿∞([−𝜏, 0]; ℝ), if 𝑥(𝑡) denotes the classical state at time 𝑡 ≥ 0, and 𝑢
+ ∈ 𝒰𝑒+
and 𝑦+∈ 𝒴𝑒+ are the future input and output signal respectively, then
( 𝑥(𝑡) 𝑅[0,𝑡]𝑦+ ) = ( 𝕋𝑡 Φ𝑡 Ψ𝑡 𝔽𝑡 ) ⋅ ( 𝑥(0) 𝑅[0,𝑡]𝑢+ ) Thus, we obtain 𝑦+(𝑡) = { 𝑥(0)(𝑡 − 𝜏), for 𝑡 ∈ [0, 𝜏]; 𝑢+(𝑡 − 𝜏), for 𝑡 > 𝜏. (2.17)
We know by comparing (2.16) and (2.17) that the initial state space 𝔅−
𝜏/ ∼ is actually
equivalent to 𝑋 = 𝐿∞([−𝜏, 0]; ℝ).
2.3. Notion of stability. Given normed signal spaces 𝒰, 𝒴 and 𝒲 ≜ 𝒰 × 𝒴, consider a system 𝑄 with initial state space 𝔖𝑄 at initial time 0 (see Definition 2.5).
Suppose that the system 𝑄 is well-posed, then we know 𝑄𝑥0is an operator from 𝒰+
𝑒 to
𝒴+
𝑒 for any 𝑥0∈ 𝔖𝑄. Moreover, if the system 𝑄 is causal, so is also 𝑄𝑥0. It is easy to
see that 𝔅𝑄= ∪𝑥0∈𝔖𝑄{𝑤−∧(𝑢+, 𝑄𝑥0𝑢+) ∣ 𝑤−∈ 𝑥0, 𝑢+∈ 𝒰𝑒+}. Thus we can regard
the system 𝑄 as a family of operators {𝑄𝑥0 : 𝑥
0∈ 𝔖𝑄} indexed by initial states.
Definition 2.11. The system 𝑄 is said to be input to output stable if and only if
it is well-posed and causal, and there exist functions2𝛽 ∈ 𝒦ℒ and 𝛾 ∈ 𝒦
∞such that, 2A function 𝛾 : [0, 𝑎) → [0, ∞) is said to be of class 𝒦 if it is continuous, strictly increasing and
satisfies 𝛾(0) = 0; moreover, if 𝑎 = ∞ and lim𝑠→∞𝛾(𝑠) = ∞, then it is said to be of class 𝒦∞. A
function 𝛽 : [0, 𝑎) × ℝ+→ [0, ∞) is said to be of class 𝒦ℒ if it is such that 𝛽(⋅, 𝑡) ∈ 𝒦 for each fixed 𝑡 ∈ ℝ+, and the function 𝛽(𝑠, ⋅) is decreasing and lim𝑡→∞𝛽(𝑠, 𝑡) = 0 for each fixed 𝑠 ∈ [0, 𝑎).
𝑃 𝐶 𝑢0 𝑢1 𝑢2 𝑦1 𝑦2 𝑦0 ? -6
Fig. 2.1. Closed-loop system [𝑃, 𝐶]
∀𝑥0∈ 𝔖𝑄, ∀𝑡 > 0, ∀𝑢0+ ∈ 𝒰+, we have ∣(𝑄𝑥0𝑢0+)(𝑡)∣ ≤ 𝛽 (𝜒(𝑥0), 𝑡) + 𝛾(∥𝑢0+∥[0,𝑡)), where the real-valued functional 𝜒(⋅) is defined by (2.6).
The above ISS-like definition represents a generalization of input-to-state stabil-ity (ISS) introduced in [25] (see also e.g., [26, 29, 31]) for the system ˙𝑥 = 𝑓(𝑥, 𝑢),
𝑦 = 𝑥 wherein the term 𝛽 (𝜒(𝑥0), 𝑡) is replaced by 𝛽 (∥𝑥0∥, 𝑡) in Sontag’s definition,
and where 𝑥0 is the initial state 𝑥0 = 𝑥(0) ∈ ℝ𝑛 rather than the abstract initial
condition developed here, which is appropriate for the more general system classes under consideration. More generally, the concept of input-to-output stability (IOS) [31] permits the more general output map 𝑦 = ℎ(𝑥).
2.4. Closed-loop systems. Consider the standard feedback configuration de-picted in Fig. 2.1 with the following equations
[𝑃, 𝐶] : 𝑤𝑖= (𝑢𝑖, 𝑦𝑖) (𝑖 = 0, 1, 2), 𝑤0= 𝑤1+ 𝑤2, (2.18)
where (𝑢0, 𝑦0) denote external disturbance; (𝑢1, 𝑦1) are the input-output pairs of the
plant 𝑃 to be controlled; and (𝑢2, 𝑦2) are the output-input pairs of the controller 𝐶.
Definition 2.12. Given normed signal spaces 𝒰, 𝒴, 𝒲 ≜ 𝒰 ×𝒴. Let the plant 𝑃
and the controller 𝐶 be represented by the sets 𝔅𝑃 and 𝔅𝐶, respectively3. We define
the closed-loop system [𝑃, 𝐶] by the following set 𝔅𝑃//𝐶 which is the interconnection
of the plant 𝑃 and controller 𝐶 shown in Fig. 2.1 that satisfies (2.18),
𝔅𝑃//𝐶 ≜ {(𝑤0, 𝑤1) ∈ 𝒲𝑒× 𝒲𝑒 ∣ 𝑤1∈ 𝔅𝑃, 𝑤2≜ 𝑤0− 𝑤1∈ 𝔅𝐶}. (2.19)
In 𝔅𝑃//𝐶 we view the external input 𝑤0 as the (closed-loop) input and the internal
signal 𝑤1as the (closed-loop) output. For the set 𝔅𝑃//𝐶, we can define the initial state
space at initial time 0 of 𝔅𝑃//𝐶 in terms of Definition 2.5, i.e., let 𝔅−𝑃//𝐶 ≜ 𝑅−𝔅𝑃//𝐶,
we similarly define an equivalence relation ∼ on 𝔅−
𝑃//𝐶 as (2.3), and the set of all
equivalence classes 𝔅−𝑃//𝐶/ ∼ is denoted as 𝔖𝑃//𝐶 which we call initial state space
of 𝔅𝑃//𝐶 at initial time 0. The size of any initial state in 𝔖𝑃//𝐶 is similarly defined
by (2.6). We next seek to establish the relationship between the initial conditions of the interconnected system, 𝔖𝑃//𝐶, the plant, 𝔖𝑃, and the controller, 𝔖𝐶.
2.4.1. Initial conditions for the closed-loop and its subsystems. For the classical state space model, it is natural to define the initial state of the closed-loop system by Cartesian product of the initial states of corresponding subsystems. In the following, we will give some answer about the relation between 𝔖𝑃//𝐶 and 𝔖𝑃× 𝔖𝐶.
Suppose that the size of any 𝑥0= (𝑥10, 𝑥20) ∈ 𝔖𝑃 × 𝔖𝐶 is defined in the usual way, 3Note that when considering the controller 𝐶, we need interchange the role of 𝒰𝑒 and 𝒴𝑒 and
e.g., for an appropriate 𝑞 ∈ [1, ∞], 𝜒(𝑥0) ≜ (𝜒(𝑥10)𝑞+ 𝜒(𝑥20)𝑞) 1 𝑞 = inf { ∥(𝑤1−, 𝑤2−)∥ 𝑤1− ∈ 𝑥10, 𝑤2−∈ 𝑥20 } . (2.20)
Note that for any 𝑠0 ∈ 𝔖𝑃//𝐶 and 𝑤0+ ∈ 𝒲𝑒+, we have defined a set Π𝑠𝑃//𝐶0 (𝑤0+)
according to (2.5) and (2.4) (let 𝔅𝑄= 𝔅𝑃//𝐶 and Π𝑠𝑃//𝐶0 (𝑤0+) = 𝑄𝑠0(𝑤0+)), i.e.,
Π𝑠0 𝑃//𝐶(𝑤0+) ≜ { 𝑤1+∈ 𝒲𝑒+ (𝑤0−, 𝑤1−∀(𝑤)∧(𝑤0+, 𝑤1+) ∈ 𝔅𝑃//𝐶, 0−, 𝑤1−) ∈ 𝑠0 } . (2.21) To understand the relation between 𝔖𝑃//𝐶 and 𝔖𝑃× 𝔖𝐶, we need to define another
set which is related to the product state 𝔖𝑃× 𝔖𝐶, denoted by Π𝑥𝑃//𝐶0 (𝑤0+), for any 𝑥0= (𝑥10, 𝑥20) ∈ 𝔖𝑃 × 𝔖𝐶 and any 𝑤0+∈ 𝒲𝑒+, as follows4:
Π𝑥0 𝑃//𝐶(𝑤0+) ≜ { 𝑤1+∈ 𝒲𝑒+ (𝑤0−∀ (𝑤, 𝑤1−)∧(𝑤0+, 𝑤1+) ∈ 𝔅𝑃//𝐶, 1−, 𝑤0−− 𝑤1−) ∈ 𝑥0 } . (2.22) Theorem 2.13. There exists a surjective and bounded5map 𝜋 : 𝔖
𝑃 × 𝔖𝐶 →
𝔖𝑃//𝐶 such that Π𝑥𝑃//𝐶0 (𝑤0+) = Π𝜋(𝑥𝑃//𝐶0)(𝑤0+) for all 𝑥0∈ 𝔖𝑃×𝔖𝐶 and all 𝑤0+∈ 𝒲𝑒+.
If we define an equivalence relation ∼ on 𝔖𝜋 𝑃 × 𝔖𝐶 by: 𝑥0 ∼ 𝑦𝜋 0 ⇔ 𝜋(𝑥0) = 𝜋(𝑦0), and the equivalence class [𝑥0] ≜ { 𝑦0 ∈ 𝔖𝑃 × 𝔖𝐶∣ 𝑦0 ∼ 𝑥𝜋 0}, and the size 𝜒([𝑥0]) ≜
inf {𝜒(𝑦0) ∣ 𝑦0∈ [𝑥0]}, and another map ¯𝜋 induced by 𝜋 as follows
¯𝜋 : (𝔖𝑃× 𝔖𝐶)/∼𝜋 → 𝔖𝑃//𝐶, ¯𝜋([𝑥0]) = 𝜋(𝑥0). (2.23)
Then ¯𝜋 is a bijective and bounded map, and the inverse ¯𝜋−1 is also bounded.
Proof. For any 𝑥0= (𝑥10, 𝑥20) ∈ 𝔖𝑃× 𝔖𝐶, choose any 𝑤1−∈ 𝑥10, 𝑤2−∈ 𝑥20and
define 𝑤0−≜ 𝑤1−+ 𝑤2−. From Definitions 2.5 and 2.12, we get 𝑠0≜ [(𝑤0−, 𝑤1−)] ∈
𝔖𝑃//𝐶. Next, we show that 𝑠0is independent of the choice of 𝑤1−∈ 𝑥10, 𝑤2−∈ 𝑥20.
Choose any other 𝑤′
1− ∈ 𝑥10, 𝑤′2− ∈ 𝑥20 and define 𝑤′0− = 𝑤1−′ + 𝑤2−′ , thus
we have 𝑠′
0 ≜ [(𝑤′0−, 𝑤1−′ )] ∈ 𝔖𝑃//𝐶. We need to show 𝑠′0 = 𝑠0. According to
(2.3) and the definition of the equivalence class (see §2.2), this is equivalent to say Π(𝑤0−,𝑤1−)
𝑃//𝐶 (𝑤0+) = Π
(𝑤′
0−,𝑤′1−)
𝑃//𝐶 (𝑤0+) for any 𝑤0+ ∈ 𝒲𝑒+. In order to prove this
equalities, by symmetry, we only need to show Π(𝑤0−,𝑤1−)
𝑃//𝐶 (𝑤0+) ⊂ Π
(𝑤′
0−,𝑤′1−)
𝑃//𝐶 (𝑤0+)
for all 𝑤0+ ∈ 𝒲𝑒+. To this end, for any 𝑤1+ ∈ Π𝑃//𝐶(𝑤0−,𝑤1−)(𝑤0+), we define 𝑤2+ = 𝑤0+− 𝑤1+, and thus from Definition 2.12 we have 𝑤1−∧𝑤1+∈ 𝔅𝑃 and 𝑤2−∧𝑤2+ ∈
𝔅𝐶. Since both 𝑤1− and 𝑤′1− belong to 𝑥10, we have from the definition of initial
conditions for 𝑃 that 𝑃𝑤1−(𝑢1+) = 𝑃𝑤′1−(𝑢1+) for all 𝑢1+ ∈ 𝒰+
𝑒. This implies that
𝑤′
1−∧𝑤1+ ∈ 𝔅𝑃. By similar argument, we also have 𝑤′2−∧𝑤2+ ∈ 𝔅𝐶. Thus, from
Definition 2.12, we obtain (𝑤′
0−∧𝑤0+, 𝑤1−′ ∧𝑤1+) ∈ 𝔅𝑃//𝐶. This implies that 𝑤1+∈
Π(𝑤′0−,𝑤′1−)
𝑃//𝐶 (𝑤0+) and thus Π(𝑤𝑃//𝐶0−,𝑤1−)(𝑤0+) ⊂ Π(𝑤 ′
0−,𝑤′1−)
𝑃//𝐶 (𝑤0+). Therefore, 𝑠0is only
related to 𝑥10and 𝑥20. We also have Π𝑥𝑃//𝐶0 (𝑤0+) = Π𝑃//𝐶𝑠0 (𝑤0+) for any 𝑤0+∈ 𝒲𝑒+. 4Note that if [𝑃, 𝐶] is well-posed (see §2.4.3 below) then Π𝑠0
𝑃//𝐶in (2.21) (resp. Π𝑥0𝑃//𝐶in (2.22))
actually defines an operator from 𝒲+
𝑒 to 𝒲𝑒+for any initial state 𝑠0∈ 𝔖𝑃//𝐶(resp. 𝑥0∈ 𝔖𝑃×𝔖𝐶).
Moreover, we have a natural surjective map 𝜋 : 𝔖𝑃× 𝔖𝐶→ 𝔖𝑃//𝐶 defined in Theorem 2.13 such
that Π𝜋(𝑥0)
𝑃//𝐶= Π𝑥0𝑃//𝐶 for any 𝑥0∈ 𝔖𝑃× 𝔖𝐶.
5Here bounded means that there exists a positive number 𝑟 ≥ 0 such that 𝜒(𝜋(𝑥0)) ≤ 𝑟 ⋅ 𝜒(𝑥0)
A natural map 𝜋 : 𝔖𝑃 × 𝔖𝐶 → 𝔖𝑃//𝐶 can be defined by 𝑥0 7→ 𝑠0. From
(2.6) and 𝑠0 = [(𝑤0−, 𝑤1−)], we have 𝜒(𝜋(𝑥0)) = 𝜒(𝑠0) ≤ ∥(𝑤0−, 𝑤1−)∥ = ∥(𝑤1−+ 𝑤2−, 𝑤1−)∥ ≤ (∥𝑤1−+ 𝑤2−∥𝑞+ ∥𝑤1−∥𝑞)1/𝑞 ≤ (2𝑞+ 1)1/𝑞(∥𝑤1−∥𝑞+ ∥𝑤2−∥𝑞)1/𝑞 for
any 𝑞 ≥ 1. Since 𝑤1− and 𝑤2− are arbitrarily chosen from 𝑥10 and 𝑥20, respectively,
we have 𝜒(𝜋(𝑥0)) ≤ (2𝑞+ 1)1/𝑞⋅ 𝜒(𝑥0). This implies that the map 𝜋 is bounded. Next
we show that 𝜋 is also a surjective map. To this end, for any 𝑠′′
0 ∈ 𝔖𝑃//𝐶, choose any
(𝑤′′
0−, 𝑤′′1−) ∈ 𝑠′′0 and define 𝑤2−′′ ≜ 𝑤0−′′ − 𝑤′′1−, thus, from (2.19) and (2.1), we have 𝑤′′
1−∈ 𝔅−𝑃 and 𝑤2−′′ ∈ 𝔅−𝐶. Define 𝑥′′10≜ [𝑤′′1−], 𝑥′′20≜ [𝑤′′2−] and 𝑥′′0 ≜ (𝑥′′10, 𝑥′′20), we
have 𝑥′′
0 ∈ 𝔖𝑃 × 𝔖𝐶 and 𝜋(𝑥′′0) = 𝑠′′0. This implies that the map 𝜋 is surjective.
Define a map ¯𝜋 by (2.23), it is easy to see that ¯𝜋 is bijective. It follows from
𝜒(¯𝜋([𝑥0])) = 𝜒(𝜋(𝑥0)) ≤ (2𝑞+ 1)1/𝑞⋅ 𝜒(𝑥0) for any 𝑞 ≥ 1 that the map ¯𝜋 is bounded.
Finally, we show that the inverse map ¯𝜋−1 : 𝔖
𝑃//𝐶 → (𝔖𝑃 × 𝔖𝐶)/𝜋
∼ is also
bounded. To this end, for any 𝑠′′
0 ∈ 𝔖𝑃//𝐶, from the proof of map 𝜋 being surjective,
we have ¯𝜋−1(𝑠′′
0) = [𝑥′′0]. Thus by applying (2.20), we get 𝜒(¯𝜋−1(𝑠′′0)) = 𝜒([𝑥′′0]) ≤ 𝜒(𝑥′′
0) ≤ (∥𝑤′′1−∥𝑞+∥𝑤2−′′ ∥𝑞)1/𝑞= (∥𝑤′′1−∥𝑞+∥𝑤0−′′ −𝑤′′1−∥𝑞)1/𝑞≤ (2𝑞+1)1/𝑞⋅(∥𝑤′′0−∥𝑞+ ∥𝑤′′
1−∥𝑞)1/𝑞. Since (𝑤′′0−, 𝑤1−′′ ) is arbitrarily chosen from 𝑠′′0, we have 𝜒(¯𝜋−1(𝑠′′0)) ≤
(2𝑞+ 1)1/𝑞⋅ 𝜒(𝑠′′
0). This implies the inverse map ¯𝜋−1 is also bounded.
2.4.2. State space initial conditions for closed-loop systems. Consider the closed-loop system shown in Fig. 2.1. The forward and feedback loop represent the plant 𝑃 and controller 𝐶, respectively. Both 𝑃 and 𝐶 with classical initial state spaces 𝑥𝑝 ∈ 𝑋𝑝 = ℝ𝑛𝑝 and 𝑥𝑐 ∈ 𝑋𝑐 = ℝ𝑛𝑐, respectively, are defined like (2.7), i.e.,
˙𝑥𝑝 = 𝑓𝑝(𝑥𝑝, 𝑢1), 𝑦1 = ℎ𝑝(𝑥𝑝, 𝑢1) and ˙𝑥𝑐 = 𝑓𝑐(𝑥𝑐, 𝑦2), 𝑢2 = ℎ𝑐(𝑥𝑐, 𝑦2). Figure 2.1
represents the following closed-loop equations:
˙𝑥𝑝= 𝑓𝑝(𝑥𝑝, 𝑢1), ˙𝑥𝑐= 𝑓𝑐(𝑥𝑐, 𝑦0− 𝑦1), (2.24a) 𝑢1= 𝑢0− ℎ𝑐(𝑥𝑐, 𝑦0− 𝑦1), 𝑦1= ℎ𝑝(𝑥𝑝, 𝑢1), (2.24b)
with product state space 𝑋𝑝× 𝑋𝑐 and (𝑢0, 𝑦0) as inputs, and (𝑢1, 𝑦1) as outputs.
With the concepts of complete, forward observable and strongly backward ob-servable defined by Definitions 2.7 and 2.8, we have the following:
Theorem 2.14. Suppose that 𝑃 , 𝐶 and the closed-loop (2.24) are complete. If
both 𝑃 and 𝐶 are forward observable (resp. strongly backward observable), then the closed-loop (2.24) is forward observable (resp. strongly backward observable).
Proof. We establish forward observability of (2.24) by contradiction. It is thus
assumed that there exist (𝑥𝑝0, 𝑥𝑐0) ∈ 𝑋𝑝× 𝑋𝑐, (𝑥′𝑝0, 𝑥′𝑐0) ∈ 𝑋𝑝× 𝑋𝑐 with (𝑥𝑝0, 𝑥𝑐0) ∕=
(𝑥′
𝑝0, 𝑥′𝑐0) such that (𝑢1, 𝑦1)∣𝑡≥0= (𝑢′1, 𝑦1′)∣𝑡≥0 for all (𝑢0, 𝑦0)∣𝑡≥0= (𝑢′0, 𝑦0′)∣𝑡≥0. This
implies that (𝑦1, 𝑢2)∣𝑡≥0= (𝑦1′, 𝑢′2)∣𝑡≥0for any (𝑢1, 𝑦2)∣𝑡≥0= (𝑢′1, 𝑦2′)∣𝑡≥0which satisfy
˙𝑥𝑝= 𝑓𝑝(𝑥𝑝, 𝑢1), ˙𝑥𝑐= 𝑓𝑐(𝑥𝑐, 𝑦2), (𝑥𝑝(0), 𝑥𝑥(0)) = (𝑥𝑝0, 𝑥𝑐0),
𝑢′
1= 𝑢1= 𝑢0− ℎ𝑐(𝑥𝑐, 𝑦2), 𝑦2′ = 𝑦2= 𝑦0− ℎ𝑝(𝑥𝑝, 𝑢1).
Since (𝑢1, 𝑦2)∣𝑡≥0 = (𝑢1′, 𝑦′2)∣𝑡≥0 can thus be taken as any element by choosing 𝑢0 = 𝑢1+ ℎ𝑐(𝑥𝑐, 𝑦2) and 𝑦0 = 𝑦2+ ℎ𝑝(𝑥𝑝, 𝑢1) with ˙𝑥𝑝 = 𝑓𝑝(𝑥𝑝, 𝑢1), ˙𝑥𝑐 = 𝑓𝑐(𝑥𝑐, 𝑦2) and
(𝑥𝑝(0), 𝑥𝑥(0)) = (𝑥𝑝0, 𝑥𝑐0), this shows that
(𝑦1, 𝑢2)∣𝑡≥0= (𝑦′1, 𝑢′2)∣𝑡≥0 for any (𝑢1, 𝑦2)∣𝑡≥0= (𝑢′1, 𝑦′2)∣𝑡≥0.
This contradicts forward observability of 𝑃 and 𝐶. Thus (2.24) is forward observable. We also show strongly backward observability of the closed-loop (2.24) by contra-diction. Assume therefore that there exist (𝑥𝑝0, 𝑥𝑐0) ∈ 𝑋𝑝× 𝑋𝑐, (𝑥′𝑝0, 𝑥′𝑐0) ∈ 𝑋𝑝× 𝑋𝑐
with (𝑥𝑝0, 𝑥𝑐0) ∕= (𝑥′𝑝0, 𝑥′𝑐0) and (𝑢0, 𝑦0)∣𝑡≤0 = (𝑢′0, 𝑦0′)∣𝑡≤0 such that (𝑢1, 𝑦1)∣𝑡≤0 =
(𝑢′
1, 𝑦1′)∣𝑡≤0, and thus (𝑢2, 𝑦2)∣𝑡≤0= (𝑢′2, 𝑦′2)∣𝑡≤0. This implies that there exist 𝑢1∣𝑡≤0=
𝑢′
1∣𝑡≤0 and 𝑦2∣𝑡≤0 = 𝑦2′∣𝑡≤0 such that 𝑦1∣𝑡≤0 = 𝑦1′∣𝑡≤0 and 𝑢2∣𝑡≤0 = 𝑢′2∣𝑡≤0. This is
a contradiction to that both 𝑃 and 𝐶 are strongly backward observable. Thus the closed-loop (2.24) is strongly backward observable.
Consider the set 𝔅𝑃//𝐶 which consists of all input-output pairs ((𝑢0, 𝑦0), (𝑢1, 𝑦1))
satisfying (2.24). Using the same procedure in §2.2, the initial state space 𝔖𝑃//𝐶 (see
Definition 2.5) for the set 𝔅𝑃//𝐶 at initial time 0 can be defined.
Theorem 2.15. Suppose that 𝑃 , 𝐶 and the closed-loop (2.24) are complete. If
both 𝑃 and 𝐶 are forward observable and strongly backward observable. Then there exists a bijective map from 𝑋𝑝× 𝑋𝑐 to 𝔖𝑃//𝐶.
Proof. This follows directly from Theorem 2.14 and Proposition 2.9.
2.4.3. Well-posedness and stability of closed-loop systems. Since the closed-loop system [𝑃, 𝐶] represented by 𝔅𝑃//𝐶 is a system in terms of Definition
2.1, we can similarly define the existence, uniqueness and well-posedness of 𝔅𝑃//𝐶
by Definition 2.4, wherein 𝑤0 ∈ 𝒲𝑒 is the input and 𝑤1 ∈ 𝒲𝑒 is the output. That
is, the closed-loop system [𝑃, 𝐶] has the existence property if for any 𝑠0 ∈ 𝔖𝑃//𝐶
and any 𝑤0+ ∈ 𝒲𝑒+ there exists a 𝑤1+ ∈ 𝒲𝑒+ such that 𝑤1+ ∈ Π𝑠𝑃//𝐶0 (𝑤0+) with
Π𝑠0
𝑃//𝐶(𝑤0+) defined by (2.21); and the uniqueness property if for all 𝑠0 ∈ 𝔖𝑃//𝐶
and all 𝑤0+ ∈ 𝒲𝑒+, 𝑤1+, ˜𝑤1+ ∈ Π𝑃//𝐶𝑠0 (𝑤0+) ⇒ 𝑤1+ = ˜𝑤1+ and is well-posed if
it has both the existence and uniqueness properties. Note that by Theorem 2.13, it also follows that 𝑠0∈ 𝔖𝑃//𝐶 and Π𝑠𝑃//𝐶0 can be replaced throughout in the above by
𝑠0∈ 𝔖𝑃 × 𝔖𝐶 and Π𝑠𝑃//𝐶0 respectively.
Following directly from Definition 2.11, we define input to output stability for the closed-loop system [𝑃, 𝐶].
Definition 2.16. The closed-loop system [𝑃, 𝐶] with initial state space 𝔖𝑃//𝐶 is
said to be input to output stable if and only if it is well-posed and causal, and there exist functions 𝛽 ∈ 𝒦ℒ and 𝛾 ∈ 𝒦∞ such that, ∀𝑠0∈ 𝔖𝑃//𝐶, ∀𝑡 > 0, ∀𝑤0+ ∈ 𝒲+,
∣(Π𝑠0
𝑃//𝐶𝑤0+)(𝑡)∣ ≤ 𝛽 (𝜒(𝑠0), 𝑡) + 𝛾(∥𝑤0+∥[0,𝑡)).
Note that two well-posed open subsystems (plant and controller) does not necessar-ily result in a well-posed closed-loop system, and that the causality of a closed-loop system doesn’t follow from the causality of open-loop subsystems (plant and con-troller) [35, §4.3.2]. The following theorem gives an alternative characterization of the property of input to output stable for a closed-loop system.
Theorem 2.17. Suppose that the closed-loop system [𝑃, 𝐶] is well-posed and
causal. The following four statements are equivalent: I. The closed-loop system [𝑃, 𝐶] is input to output stable.
II. There exist 𝛽1∈ 𝒦ℒ and 𝛾1∈ 𝒦∞such that, ∀𝑠0∈ 𝔖𝑃//𝐶, ∀𝑡 > 0, ∀𝑤0+∈ 𝒲+, ∣(Π𝑠0
𝑃//𝐶𝑤0+)(𝑡)∣ ≤ 𝛽1(𝜒(𝑠0), 𝑡) + 𝛾1(∥𝑤0+∥[0,𝑡)). (2.25) III. There exist 𝛽2∈ 𝒦ℒ, 𝛾2∈ 𝒦∞such that, ∀𝑥0∈ 𝔖𝑃×𝔖𝐶, ∀𝑡 > 0, ∀𝑤0+∈ 𝒲+,
∣(Π𝑥0
𝑃//𝐶𝑤0+)(𝑡)∣ ≤ 𝛽2(𝜒(𝑥0), 𝑡) + 𝛾2(∥𝑤0+∥[0,𝑡)). (2.26) IV. There exist 𝛽3∈ 𝒦ℒ and 𝛾3∈ 𝒦∞such that, ∀𝑥0= (𝑥10, 𝑥20) ∈ 𝔖𝑃× 𝔖𝐶, ∀𝑡 >
0, ∀𝑤0+∈ 𝒲+, ∀𝑤1−∈ 𝑥10, ∀𝑤2−∈ 𝑥20, ∣(Π𝑥0
Moreover, we have 𝛾1= 𝛾2= 𝛾3 and 𝛽2= 𝛽3. Proof. I ⇔ II: This follows from Definition 2.16.
II ⇒ III: Suppose that (2.25) holds with given functions 𝛽1∈ 𝒦ℒ, 𝛾1∈ 𝒦∞. For
any 𝑥0∈ 𝔖𝑃× 𝔖𝐶, by Theorem 2.13, we have 𝜋(𝑥0) ∈ 𝔖𝑃//𝐶 and Π𝑥𝑃//𝐶0 = Π𝜋(𝑥𝑃//𝐶0),
and 𝜒(𝜋(𝑥0)) ≤ ∥𝜋∥⋅𝜒(𝑥0) (note that 𝜋 is a bounded map). Define a function 𝛽2∈ 𝒦ℒ
by 𝛽2(𝑟, 𝑡) ≜ 𝛽1(∥𝜋∥𝑟, 𝑡) for all 𝑟 ≥ 0, 𝑡 ≥ 0. We have (2.26) holds with 𝛾2= 𝛾1.
III ⇒ II: Suppose that (2.26) holds with given functions 𝛽2 ∈ 𝒦ℒ and 𝛾2 ∈ 𝒦∞. For any 𝑠0 ∈ 𝔖𝑃//𝐶, by Theorem 2.13, we have ¯𝜋−1(𝑠0) ∈ (𝔖𝑃 × 𝔖𝐶)/∼𝜋 and
𝜒(¯𝜋−1(𝑠0)) ≤ ∥¯𝜋−1∥𝜒(𝑠0) (note that ¯𝜋−1is a bounded bijective map). For any 𝜀 > 0,
there exists a 𝑥0 ∈ 𝔖𝑃 × 𝔖𝐶 such that 𝑥0 ∈ ¯𝜋−1(𝑠0) and 𝜒(𝑥0) ≤ 𝜒(¯𝜋−1(𝑠0)) + 𝜀.
Thus we have ∣(Π𝑠0
𝑃//𝐶𝑤0+)(𝑡)∣ = ∣(Π𝑥𝑃//𝐶0 𝑤0+)(𝑡)∣ ≤ 𝛽2(𝜒(𝑥0), 𝑡) + 𝛾2(∥𝑤0+∥[0,𝑡)) ≤ 𝛽2(∥¯𝜋−1∥ ⋅ 𝜒(𝑠0) + 𝜀, 𝑡)+ 𝛾2(∥𝑤0+∥[0,𝑡)) for any 𝑡 > 0 and any 𝑤0+ ∈ 𝒲+. Since 𝜀 is an arbitrarily chosen positive number, we have (2.25) holds with 𝛾1 = 𝛾2 and 𝛽1(𝑟, 𝑡) = 𝛽2(∥¯𝜋−1∥ ⋅ 𝑟, 𝑡) for all 𝑟 ≥ 0 and 𝑡 ≥ 0.
III ⇒ IV: Suppose that (2.26) holds with given functions 𝛽2∈ 𝒦ℒ and 𝛾2∈ 𝒦∞.
From (2.20), we know that 𝜒(𝑥0) ≤ ∥(𝑤1−, 𝑤2−)∥ for any 𝑤1−∈ 𝑥10 and any 𝑤2− ∈ 𝑥20. Thus, we have (2.27) holds with 𝛽3= 𝛽2 and 𝛾3= 𝛾2.
IV ⇒ III: Suppose that (2.27) holds with given functions 𝛽3∈ 𝒦ℒ and 𝛾3∈ 𝒦∞.
For any 𝑥0 = (𝑥10, 𝑥20) ∈ 𝔖𝑃 × 𝔖𝐶, for any 𝜀 > 0, from (2.20), we know that
there exist 𝑤1− ∈ 𝑥10 and 𝑤2− ∈ 𝑥20 such that ∥(𝑤1−, 𝑤2−)∥ ≤ 𝜒(𝑥0) + 𝜀. Thus
we have ∣(Π𝑥0
𝑃//𝐶𝑤0+)(𝑡)∣ ≤ 𝛽3(∥(𝑤1−, 𝑤2−)∥, 𝑡) + 𝛾3(∥𝑤0+∥[0,𝑡)) ≤ 𝛽3(𝜒(𝑥0) + 𝜀, 𝑡) + 𝛾3(∥𝑤0+∥[0,𝑡)) for all 𝑡 ≥ 0 and all 𝑤0+∈ 𝒲+. Since 𝜀 is an arbitrarily chosen positive
number, we have (2.26) holds with 𝛽2= 𝛽3 and 𝛾2= 𝛾3.
3. Robust stability and main results. Given normed signal spaces 𝒰, 𝒴 and
𝒲 ≜ 𝒰 ×𝒴, consider the closed-loop system [𝑃, 𝐶] with the plant 𝑃 and the controller 𝐶 (Definition 2.12). Let the perturbed plant ˜𝑃 and the perturbed closed-loop system
[ ˜𝑃 , 𝐶] be represented by the sets 𝔅𝑃˜ ⊂ 𝒰𝑒× 𝒴𝑒and 𝔅𝑃 //𝐶˜ ⊂ 𝒲𝑒× 𝒲𝑒, respectively.
Let 𝔖𝑃, 𝔖𝑃˜, 𝔖𝐶, 𝔖𝑃//𝐶 and 𝔖𝑃 //𝐶˜ be the corresponding initial state spaces of 𝔅𝑃,
𝔅𝑃˜, 𝔅𝐶, 𝔅𝑃//𝐶 and 𝔅𝑃 //𝐶˜ at initial time 0, respectively. Note that, the graph 𝒢𝑤1−
𝑃 , 𝒢𝑤𝑃˜˜1− and 𝒢𝐶𝑤2− for any 𝑤1− ∈ 𝔅−𝑃, any ˜𝑤1− ∈ 𝔅−𝑃˜ and any 𝑤2− ∈ 𝔅−𝐶 are
similarly defined according to (2.2).
3.1. General systems. The main result of the paper is given next. It forms a direct extension of [13, Theorems 1 and 6] to include non-zero initial conditions. Before giving the result, we recall that an operator Ψ : 𝒲+ → 𝒲+ is said to be relatively continuous if, for all operators Φ : 𝒲+ → 𝒲+ with 𝑅
[0,𝜏)Φ compact for
any 𝜏 > 0, the operator 𝑅[0,𝜏)(Φ ∘ Ψ) : 𝒲+→ 𝒲[0, 𝜏) is continuous.
Theorem 3.1. Assume that 𝑃 , ˜𝑃 and 𝐶 are well-posed and causal systems, and that [𝑃, 𝐶] is time-invariant, well-posed and causal, and that [ ˜𝑃 , 𝐶] is causal. Let
[𝑃, 𝐶] be input to output stable, i.e., there exist functions 𝛽 ∈ 𝒦ℒ and 𝛾 ∈ 𝒦∞ such
that, ∀𝑥0= (𝑥10, 𝑥20) ∈ 𝔖𝑃× 𝔖𝐶, ∀𝑤0+∈ 𝒲+, ∀𝑡 > 0, ∣(Π𝑥0
𝑃//𝐶𝑤0+)(𝑡)∣ ≤ 𝛽 (𝜒(𝑥0), 𝑡) + 𝛾(∥𝑤0+∥[0,𝑡)). (3.1)
If there exist functions 𝜎0, 𝜎 ∈ 𝒦∞ and 𝛽0∈ 𝒦ℒ such that for any ˜𝑤1− ∈ 𝒲−∩ 𝔅−𝑃˜ there exists a 𝑤1−∈ 𝒲−∩ 𝔅−𝑃 with
and a causal surjective operator Φ : dom(Φ) ⊂ 𝒢𝑤1−
𝑃 → 𝒢𝑃𝑤˜˜1− satisfying, ∀𝑡 > ℎ ≥
0, ∀𝑤1+∈ dom(Φ),
∣((Φ − 𝐼)𝑤1+)(𝑡)∣ ≤ 𝛽0(∥𝑤1−∧𝑤1+∥(−∞,ℎ], 𝑡 − ℎ) + 𝜎(∥𝑤1+∥[ℎ,𝑡)). (3.3) In addition, if there exist two functions 𝜌, 𝜀 of class 𝒦∞ such that, ∀𝑠 ≥ 0,
𝜎 ∘ (𝐼 + 𝜌) ∘ 𝛾(𝑠) ≤ (𝐼 + 𝜀)−1(𝑠), (3.4)
and either of the following conditions is satisfied: I. [ ˜𝑃 , 𝐶] is well-posed and Π˜𝑥0
˜
𝑃 //𝐶(𝒲+) ⊂ 𝒲+ for any ˜𝑥0∈ 𝔖𝑃˜× 𝔖𝐶;
II. [ ˜𝑃 , 𝐶] has the uniqueness property, and Π𝑥0
𝑃//𝐶 is relatively continuous for any
𝑥0∈ 𝔖𝑃× 𝔖𝐶, and 𝑅[0,𝜏)(Φ − 𝐼) is compact for any 𝜏 > 0;
then the closed-loop system [ ˜𝑃 , 𝐶] is also input to output stable. More specifically, for any function 𝛼 of class 𝒦∞, there exists a function ˜𝛽 ∈ 𝒦ℒ such that, ∀˜𝑥0 ∈
𝔖𝑃˜× 𝔖𝐶, ∀ ˜𝑤0+∈ 𝒲+, ∀𝑡 > 0, ∣(Π˜𝑥0 ˜ 𝑃 //𝐶𝑤˜0+)(𝑡)∣ ≤ ˜𝛽 (𝜒(˜𝑥0), 𝑡) + (𝛼 + ˜𝛾)(∥ ˜𝑤0+∥[0,𝑡)), (3.5) where ˜𝛾 ∈ 𝒦∞ is defined by ˜𝛾(𝑟) ≜ (𝜎 + 𝐼) ∘ (𝐼 + 𝜌) ∘ 𝛾 ∘ (𝐼 + 𝜀−1)3(𝑟), ∀𝑟 ≥ 0. (3.6) Proof. (Part I) For any ˜𝑤0+ ∈ 𝒲+ and any ˜𝑥0 ∈ 𝔖𝑃˜ × 𝔖𝐶, choose any
bounded ( ˜𝑤1−, 𝑤2−) ∈ ˜𝑥0 and let ˜𝑤0− = ˜𝑤1− + 𝑤2−. Since [ ˜𝑃 , 𝐶] is well-posed,
causal and Π˜𝑥0 ˜
𝑃 //𝐶(𝒲+) ⊂ 𝒲+, there exists a unique ( ˜𝑤1+, 𝑤2+) ∈ 𝒲+ × 𝒲+
such that ˜𝑤1+ ∈ 𝒢𝑃𝑤˜˜1−, 𝑤2+ ∈ 𝒢𝐶𝑤2− and ˜𝑤0+ = ˜𝑤1+ + 𝑤2+, i.e., the operator
Π˜𝑥0 ˜
𝑃 //𝐶 : 𝒲+→ 𝒲+, ˜𝑤0+7→ ˜𝑤1+ is well defined and causal.
Under conditions in Theorem 3.1, there exists a 𝑤1− ∈ 𝒲−∩ 𝔅−𝑃 for ˜𝑤1− such
that ∥𝑤1−∥ ≤ 𝜎0(∥ ˜𝑤1−∥) (see (3.2)), and thus
∥(𝑤1−, 𝑤2−)∥ ≤ (𝜎0+ 𝐼)(∥( ˜𝑤1−, 𝑤2−)∥). (3.7)
In addition, there exists a causal surjective operator Φ : 𝒢𝑤1−
𝑃 → 𝒢𝑤𝑃˜˜1−. It follows from
the surjection of Φ that there exists 𝑤1+∈ 𝒢𝑃𝑤1− satisfying Φ(𝑤1+) = ˜𝑤1+. We choose 𝑥0 ≜ ([𝑤1−], [𝑤2−]) ∈ 𝔖𝑃 × 𝔖𝐶 and let 𝑤0− = 𝑤1−+ 𝑤2− and 𝑤0+ ≜ 𝑤1++ 𝑤2+.
It follows from the well-posedness of [𝑃, 𝐶] that Π𝑥0
𝑃//𝐶(𝑤0+) = 𝑤1+; and thus the
following equations hold: Π˜𝑥0 ˜ 𝑃 //𝐶( ˜𝑤0+) = ˜𝑤1+= Φ ∘ Π𝑥𝑃//𝐶0 (𝑤0+), (3.8) ˜ 𝑤0+= ( 𝐼 + (Φ − 𝐼) ∘ Π𝑥0 𝑃//𝐶 ) (𝑤0+). (3.9)
For ease of notation, we define 𝑤𝑖 ≜ (𝑤𝑖−∧𝑤𝑖+) for 𝑖 = 0, 1, 2 and ˜𝑤𝑗 ≜ ( ˜𝑤𝑗−∧ ˜𝑤𝑗+)
for 𝑗 = 0, 1. From equation (3.1), Theorem 2.17 and using the time-invariance and causality of [𝑃, 𝐶], we have
Note that, for any function 𝜇 : [0, 𝑟) → [0, ∞) of class 𝒦, any function 𝜈 of class 𝒦∞
and any 𝑎 ≥ 0, 𝑏 ≥ 0 with 𝑎 + 𝑏 < 𝑟, we have6
𝜇(𝑎 + 𝑏) ≤ 𝜇 ∘ (𝐼 + 𝜈)(𝑎) + 𝜇 ∘ (𝐼 + 𝜈−1)(𝑏). (3.11)
Next, we estimate the upper bound of ∥(𝑤1, 𝑤2)∥ by first giving the upper bound of ∥𝑤0+∥. It follows from (3.9) that:
∥𝑤0+∥ ≤ ∥ ˜𝑤0+∥ + ∥(𝐼 − Φ)(Π𝑥𝑃//𝐶0 𝑤0+)∥ ≤ ∥ ˜𝑤0+∥ + 𝛽0(∥𝑤1−∥, 0) + 𝜎(∥Π𝑥𝑃//𝐶0 𝑤0+∥), [by (3.3)] ≤ ∥ ˜𝑤0+∥ + 𝛽0(∥𝑤1−∥, 0) + 𝜎(𝛽(∥(𝑤1−, 𝑤2−)∥, 0) + 𝛾(∥𝑤0+∥)), [by (3.10)] ≤ ∥ ˜𝑤0+∥ + 𝛽0((𝜎0+ 𝐼)(∥( ˜𝑤1−, 𝑤2−)∥), 0)+ 𝜎 ∘ (𝐼 + 𝜌) ∘ 𝛾(∥𝑤0+∥) + 𝜎 ∘ (𝐼 + 𝜌−1) ∘ 𝛽((𝜎 0+ 𝐼)(∥( ˜𝑤1−, 𝑤2−)∥), 0), [by (3.7) and (3.11)]
Since condition (3.4) is satisfied and (𝐼 − (𝐼 + 𝜀)−1)−1(⋅) = (𝐼 + 𝜀−1)(⋅), we see that
∥𝑤0+∥ ≤ (𝐼 + 𝜀−1)(∥ ˜𝑤0+∥ + Δ(∥( ˜𝑤1−, 𝑤2−)∥)), (3.12)
where function Δ ∈ 𝒦 is defined by:
Δ(𝑟) ≜ 𝛽0((𝜎0+ 𝐼)(𝑟), 0)+ 𝜎 ∘ (𝐼 + 𝜌−1) ∘ 𝛽((𝜎0+ 𝐼)(𝑟), 0), ∀𝑟 ≥ 0. (3.13)
Define three functions 𝛼𝑖∈ 𝒦∞, (𝑖 = 1, 2, 3) by
𝛼1(𝑠) ≜ (𝜎0+ 𝐼)(𝑠) + 2𝛽((𝜎0+ 𝐼)(𝑠), 0), ∀𝑠 ≥ 0; 𝛼2(𝑠) ≜ 𝛼1(𝑠) + (2𝛾 + 𝐼) ∘ (𝐼 + 𝜀−1) ∘ (𝐼 + 𝜀) ∘ Δ(𝑠), ∀𝑠 ≥ 0; 𝛼3(𝑠) ≜ (2𝛾 + 𝐼) ∘ (𝐼 + 𝜀−1) ∘ (𝐼 + 𝜀−1)(𝑠), ∀𝑠 ≥ 0. Thus, we have ∥(𝑤1, 𝑤2)∥ ≤ ∥(𝑤1−, 𝑤2−)∥ + 2∥𝑤1+∥ + ∥𝑤0+∥ ≤ (𝜎0+ 𝐼)(∥( ˜𝑤1−, 𝑤2−)∥) + 2𝛽((𝜎0+ 𝐼)(∥( ˜𝑤1−, 𝑤2−)∥), 0) + 2𝛾(∥𝑤0+∥) + ∥𝑤0+∥, [by (3.7) and (3.10)] (3.14) ≤ 𝛼1(∥( ˜𝑤1−, 𝑤2−)∥), [by (3.12)] + (2𝛾 + 𝐼) ∘ (𝐼 + 𝜀−1)(∥ ˜𝑤 0+∥ + Δ(∥( ˜𝑤1−, 𝑤2−)∥)) ≤ 𝛼2(∥( ˜𝑤1−, 𝑤2−)∥) + 𝛼3(∥ ˜𝑤0+∥) ≜ 𝑠∞, [by (3.11)]
By using the equation (3.9), for any 𝑡 > 0, we have
∣𝑤0+(𝑡)∣ ≤ ∣ ˜𝑤0+(𝑡)∣ + ∣((Φ − 𝐼) ∘ Π𝑥𝑃//𝐶0 (𝑤0+))(𝑡)∣ ≤ ∥ ˜𝑤0∥[0,𝑡)+ 𝛽0(∥𝑤1∥(−∞,𝑡/2], 𝑡 − 𝑡/2) + 𝜎(∥𝑤1+∥[𝑡/2,𝑡]), [by (3.3)] ≤ ∥ ˜𝑤0∥[0,𝑡)+ 𝛽0(𝑠∞, 𝑡/2) + 𝜎(𝛽(𝑠∞, 𝑡/4) + 𝛾(∥𝑤0∥[𝑡/4,𝑡))), [by (3.10)] ≤ ∥ ˜𝑤0∥[0,𝑡)+ 𝛽0(𝑠∞, 𝑡/2) + 𝜎 ∘ (𝐼 + 𝜌−1) ∘ 𝛽(𝑠∞, 𝑡/4) + 𝜎 ∘ (𝐼 + 𝜌) ∘ 𝛾(∥𝑤0∥[𝑡/4,𝑡)), [by (3.11)] ≤ ∥ ˜𝑤0∥[0,𝑡)+ 𝛽1(𝑠∞, 𝑡) + (𝐼 + 𝜀)−1(∥𝑤0+∥[𝑡/4,𝑡)), (3.15) 6if 𝑏 ≤ 𝜈(𝑎) then 𝜇(𝑎 + 𝑏) ≤ 𝜇 ∘ (𝐼 + 𝜈)(𝑎); and if 𝑎 ≤ 𝜈−1(𝑏) then 𝜇(𝑎 + 𝑏) ≤ 𝜇 ∘ (𝐼 + 𝜈−1)(𝑏).
